Here is the full list of all 14 Battle Chef Brigade Deluxe trophies - there are 11 bronze, 2 silver and 1 gold.
Use Thrash's Heavy Attack to smash a monster into 3 others
Join the Brigade as Mina
Win the practice match against Thrash
Complete all of Thorn's Hunts
Complete all of Belchior's Puzzles
Complete all of Pontida's Jobs
Beat the game without losing a duel
Win a round with over 2 minutes remaining on the match clock
Win a duel without using the theme ingredient
Make a dish worth 450 points or more
Reach a combo multiplier of 10
Kill 3 enemies without touching the ground
Complete Story Mode
Complete Story Mode on Hard